Multiple Yield Curve Modelling with CBI Processes

We develop a modelling framework for multiple yield curves driven by continuous-state branching processes with immigration (CBI processes). Exploiting the self-exciting behavior of CBI jump processes, this approach can reproduce the relevant empirical features of spreads between different interbank rates. We provide a complete analytical framework, including a detailed study of discounted exponential moments of CBI processes. The proposed framework yields explicit valuation formulae for all linear interest rate derivatives as well as semi-closed formulae for non- linear derivatives via Fourier techniques and quantization. We show that a simple specification of the model can be successfully calibrated to market data

Limit theorems for continuous-state branching processes with immigration

AbstractA continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether \int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ . When \int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).</jats:p

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